Tricki
a repository of mathematical know-how

Simplify your problem by generalizing it

Quick description

One might think that generalizing a problem would make it harder to solve. However, the reverse is often true. The articles linked to from here explain why.

General discussion

Clarify your problem by making it more abstract Quick description ( If you are asked to prove that a mathematical object X has a property P, you will often obtain a clearer and easier problem if you identify certain properties P_1,\dots,P_k of X and then prove the statement "Every object that has properties P_1,\dots,P_k has property P" instead of the original statement "X has property P." See also Think axiomatically even about concrete objects. )

Strengthen your inductive hypothesis Quick description ( Suppose that you are trying to prove by induction that a statement P(n) is true for every n. If it is hard to deduce P(n+1) from P(n) then you may be able to deal with the difficulty by finding a different statement Q(n) that implies P(n), and deducing Q(n+1) from Q(n) instead (and, of course, proving Q(1)). )

To understand an object, consider treating it as one of a family of objects and analysing the family Quick description ( An insight that has shaped much of modern mathematics is that it is often better to look at families of mathematical objects rather than at individual objects in isolation. If the family itself has a structure (usually geometric or algebraic) then it may be that this can be exploited: to prove a fact about one of the objects X in the family it may be easier to prove a result about all the objects in the family and deduce from it what you want to know about X.)